Is it possible to use physics or other form of non-canonical reasoning to study functions? It is well-known (see, for example, the books New Horizons in geometry, Maxima and minima without calculus and The Mathematical Mechanic) that it is possible to use some forms of "physical reasoning", "geometric reasoning", or "probabilistic reasoning" or otherwise non-canonical arguments to find minima and maxima of some functions or to solve some problems that normally require calculus.

Are there such methods to calculate limits, find derivatives, or
  verify the continuity of a function? Could you provide some examples?

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$Using idealized operational amplifiers (opamps), you can create idealized integrating circuits:

If the voltage $V_\text{in} - \text{Ground}$ is given by the function $V_\text{in}(t)$, and the output voltage $V_\text{out} - \text{Ground}$ is given by the function $V_\text{out}(t)$, then the circuit maintains:
$$V_\text{out}(t) = \dint{V_\text{in}(t')}{t'}{-\infty}{t}$$
as well as integrating circuits:

Similarly, if the voltage $V_\text{in} - \text{Ground}$ is given by the function $V_\text{in}(t)$, and the output voltage $V_\text{out} - \text{Ground}$ is given by the function $V_\text{out}(t)$, then the circuit maintains:
$$V_\text{out}(t) = \pred{V_\text{in}(t)}{t}$$
The main problem with this sort of "physics" approach to differential calculus is that the circuit components are idealized.  The opamps have their own power supply, so the calculation only works within a certain range.  The capacitors will explode if you try to push too much through them.  
Also the differentiator circuit is very unreliable, any small white noise can cause large spikes (effectively a momentary dirac delta distribution) on the output.  But for your question, this might be a feature, since spikes would indicate discontinuity.

Both images are from wikimedia commons.  
A: Soap films create a surface with minimal surface area naturally. There is some more at this site. Here is a video. Note that these are local rather than global minima.
